The symmetry of the superconducting order parameter within the $t-t'$ one-band model on the square lattice

Since the discovery of high-temperature superconductivity in doped cuprates a huge amount of data is obtained, which are results of theoretical and experimental studies. Despite some apparent success in high-temperature superconductivity research, still an open question remains about the mechanism of this phenomenon.
An important issue in the study of the mechanism of superconductivity in quasi-two-dimensional structures is the formation of superconducting states with different symmetries of the order parameter and their competition. There are a lot of experimental and theoretical data that provide a necessity of further studying of the competition between superconducting states with $s$- and $d$-wave pairing and the formation of state with mixed $s+id$-symmetry.
It is believed that cuprates are characterized by $d$-wave pairing [4, 5]. However, Raman spectroscopy experiments in Nd$_{2-x}$Ce$_x$CuO$_4$ [6] and ARPES experiments in La$_{2-x}$Sr$_x$CuO$_4$ [7] showed that the symmetry of the order parameter in copper oxides deviates from the traditional d-wave symmetry. Also it was concluded in experimental studies [8–10] and in the theoretical study [11] that there is a possibility of the formation of mixed $s+d$ and $s+id$-wave superconducting states in the layered cuprate systems.
There was observed the order parameter with the $s\pm$-wave symmetry [12–14] in iron-based superconductors. On the other hand, calculations which take into account spin fluctuations have shown that $d$-wave pairing may be formed in pnictides [15]. It provides a motivation to study a low-temperature transition between $s$- and $d$-wave states with the formation of intermediate phase with mixed $s+id$-wave pairing [11 16, 17].
In our investigation we use the Hartree–Fock approximation of the single-band model on the square lattice. The attraction of electrons located on the nearest-neighbour sites is taken into account. The order parameter with $s$-, $d$- and $s+id$-symmetries is considered. We have left beyond the scope of our investigation a mechanism of an electron attraction.
The ground state phase diagram of the model in the variables $t'/t$ and $n$ was constructed. It is shown that both superconducting states and the intermediate phase $s+id$ as well are realized in a wide range of the model parameters.
Results of our calculations show the formation of the superconducting state with the mixed $s+id$-wave pairing can be described in terms of a simple mean-field approximation without considering of the spin fluctuations. In this case the resulting phase diagram of superconducting states is qualitatively consistent with diagrams calculated using more sophisticated approximations and taking into account spin-fluctuations. This indicates an insignificant role of dynamic effects in the competition between various states of the superconductivity and suggests that the considered model can be used further for temperature-dependent calculations.